One central characteristic of complex social-technological systems is their self-organized emerging interactions which provide the benefits of exchanging information and resources effectively, yet at the same time increases the risk and pace of spreading attacks or failures. A small perturbation on a complex system operating in a high-risk unstable region can induce a critical transition that leads to unstoppable catastrophic failures.
There are a few recent works of early warning signals for heterogeneously networked dynamical systems. Kwon and Yang (see the List of Incorporated Cited Literature References, Literature Reference No. 1) use transfer entropy to analyze financial market data. They calculate pair-wise transfer entropy to form a transfer entropy matrix and demonstrate the asymmetrical influence from mature markets to emerging markets using transfer entropy. Their method estimates global transfer entropy, not local transfer entropy changing in time; therefore, their method cannot handle dynamics and structure changes in directional influence.
Staniek and Lehnertz (see Literature Reference No. 4) introduced a robust and computationally efficient method to estimate transfer entropy using a symbolization technique. Their demonstration of symbolic transfer entropy (STE) on brain electrical activity data shows STE is able to detect the asymmetric dependences and identify the hemisphere containing the epileptic focus without observing actual seizure activity. However, their emphasis is also not on identifying the dynamics of the complex system structure.
Scheffer et al. (see Literature Reference No. 5) surveys state-of-the-art methods that originated in the ecological domain. These methods focus only on homogeneous lattices with signals such as increased temporal correlation, skewness, and spatial correlations of the population dynamics, thus are not able to deal with heterogeneously networked dynamical systems.
Moon and Lu (see Literature References No. 2 and 3) developed a spectral early warning signals (EWS) theory that detects the approaching of critical transitions and can estimate the system structure and network connectivity near critical transitions using the covariance matrix. Although spectral EWS quantifies how much entities of a system are moving together, the symmetric nature of covariance spectrum does not permit the analysis of directional influences among entities.
Thus, a continuing need exists for a system for automated detection of critical transitions in heterogeneously networked dynamical system using an information dynamic spectrum framework that permits analysis of directional influences among entities.